Give me example of sequence of continous real valued function defined on $[0,\infty)$. Suppose $f_n(x) \rightarrow f(x)$ for all x $\in [0,\infty)$ such that

  • If $f_n \rightarrow f$ uniformly on $[0,\infty)$, then $\int\limits_0^{\infty}f_n(x) dx \nrightarrow \int\limits_0^{\infty}f(x) dx$

  • If $\int \limits_0^1 |f_n(x) - f(x) | dx \rightarrow 0$, then $f_n \nrightarrow f$ uniformly on $[0,1]$

Thank you .


Hints (the following functions are not continuous, but they show basic ideas, I'm sure you can fix the continuity):

(1) Let $f_n = \frac 1n\chi_{[0,n]}$ ($\chi_A$ denoting the characteristic function of $A$, being 1 on $A$ and $0$ on $A^c$). Then $f_n \to 0$ uniformly, but $\int_0^\infty f_n = 1$.

(2) Let $f_n = \chi_{[n,n+\frac 1n]}$. Then $f_n \to 0$, $\int_0^\infty f_n = \frac 1n$, but $\|f_n\|_\infty = 1 \not\to 0$.

  • $\begingroup$ @ martin : any example of sequence of continous function $\endgroup$ – Struggler Jun 11 '14 at 14:06
  • $\begingroup$ Thanks for your prompt reply $\endgroup$ – Struggler Jun 11 '14 at 14:06

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